Discovering Intrinsic PK/PD Models Using Physics Informed Neural Networks

Abstract

Introduction: Pharmacokinetic-pharmacodynamic (PK/PD) models employ ordinary differential equations (ODEs) for describing the relationship between dose, concentration, intensity, and response duration [1, 2, 3, 4]. These models are integral for model-informed drug discovery and development. While PK/PD models are informed by prior knowledge (i.e. biological system, mechanism of action) and data, the landscape is evolving with the rise of large datasets and the growing use of artificial intelligence (AI), with a keen interest in purely data driven approaches to model discovery. Physics informed neural networks (PINNs) have shown great promise in approximating differential equations by incorporating prior knowledge about the physics into the neural networks (NN) architecture [5, 6, 7]. A limiting assumption used in the standard PINN studies is that the functional form of the differential equation is known, which is a particular issue for problems requiring the identification of relevant models. To address this issue, we present a novel Pharamacokinetic informed neural network architecture named PKINNs which combines PINNs and Symbolic Regression methods, enabling the network to possess important features of interpretability and generalizability, to discover intrinsic mechanistic models from noisy data in addition to estimating several unknown parameters. Objectives: Determine whether PKINNs is able to discover the canonical pharmacokinetic two-compartment PK model with first-order absorption and first-order elimination process from noisy data. Investigate whether PKINNs is able to capture the intrinsic dynamics of for a model of Target-mediated drug disposition (TMDD) Methods: We generate a synthetic dataset for both the i) two-compartment model ii) TMDD model. To incorporate variability, we add Gaussian noise of different strengths to the simulated data. We choose three levels of Gaussian noise which yields three unique datasets for each model: low noise, medium noise and high noise respectively. We subset each dataset into training and test data to train the PKINNs model. We implement PKINNs and train on each dataset and use the test datasets for investigating the extrapolative capabilities of PKINNs. We employ two different methods for Symbolic regression, namely Sparse identification of nonlinear dynamics (SINDY) [8] and a genetic algorithm based model PySR [9] to probe the explainability of PKINNs. Results: PKINNs can accurately suggest structural models adequately describing the drug concentration time profiles. Interestingly, PKINNs is robust to the noise in the raw data irrespective of the noise strength which is demonstrated by the ability extract the smooth drug concentration curves which are intrinsic to the two-compartment PK model. On the other hand, PKINNs performs worse approximating the more complex TMDD model compared the two-compartment PK model. Conclusion: In this work we have developed a novel pharmacokinetic informed neural network model called PKINNs. We demonstrate that PKINNs accurately and robustly predicts the intrinsic derivatives of the underlying PK model and performs well in extrapolation predictions scenarios for relatively simple PK system. We observe that PKINNs struggles to approximate the more complex TMDD model but still provides key insights into the intrinsic model in an interpolation setting. Further exploration of the ability of PKINNs in discovering the intrinsic PK/PD structural model is warranted. This data-driven approach can serves as a valuable starting point for explainable neural network application.